### 金融代写|利率建模代写Interest Rate Modeling代考|MATH5985

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|利率建模代写Interest Rate Modeling代考|Heath-Jarrow-Morton Framework

We introduced a bond market $\mathcal{B}$ in Chapter 3 that did not admit a term structure of interest rates. If, instead, we assume a term structure in the bond market, it becomes possible to relate the bond price to the interest rate. We represent the dynamics of bond price by a stochastic process and use this to specify the corresponding interest rates. Such a system is referred to as a term structure model of interest rates (or put simply, an interest rate model).
A model specified by the dynamics of a short rate is referred to as a short rate model. A model specified by the dynamics of forward rates is referred to as a forward rate model. For management of interest rate risk, it is better to suppose various types of changes in the yield curve, and specifically to suppose changes in the forward rates. Because of this, the forward rate model is more useful in risk management than the short rate model is.

This section briefly introduces the HJM model, which is the most general forward rate model. For additional details, readers are recommended to consult Cairns (2004), Munk (2011), or Shreve (2004), among others. For details on calibration, readers are recommended to consult Wu (2009).
Forward rate process
Let $\left(\Omega, \mathcal{F}{t \in[0, \tau]}, \mathbf{P}\right)$ be a filtered probability space, where $\mathcal{F}{t \in[0, \tau]}$ is the augmented filtration and $\mathbf{P}$ denotes the real-world measure. The instantaneous forward rate with maturity $T$ observed at time $t$ is denoted by $f(t, T)$. When the usage is unambiguous, $f(t, T)$ will be called the forward rate. Typically $f(0, T)$ represents an initial forward rate.

We assume that the dynamics of $f(t, T)$ on $\left(\Omega, \mathcal{F}{t \in[0, \tau]}, \mathbf{P}\right)$ is represented by $$d f(t, T)=\alpha(t, T) d t+\sigma(t, T) d W{t},$$
where $W_{t}=\left(W_{t}^{1}, \cdots, W_{t}^{d}\right)^{T}$ is a $d$-dimensional P-Brownian motion, and $\alpha(t, T)$ and $\sigma(t, T)$ are predictable processes satisfying some technical conditions. Here, $\sigma(t, T)=\left(\sigma^{1}(t, T), \cdots, \sigma^{d}(t, T)\right)^{T}$ is a $d$-dimensional process. The second term, $\sigma(t, T) d W_{t}$ denotes the inner product of $\sigma(t, T)$ and $d W_{t}$ in $\mathbf{R}^{d}$, specifically
$$\sigma(t, T) d W_{t}=\sum_{l=1}^{d} \sigma^{l}(t, T) d W_{t}^{l} .$$

## 金融代写|利率建模代写Interest Rate Modeling代考|Arbitrage Pricing and Market Price of Risk

This section briefly studies some fundamental subjects in the HJM model, specifically, forward rate process, arbitrage pricing, the market price of risk, and state price deflator.
Forward rate process
Here let us represent the forward rate process under the risk-neutral measure Q. Differentiating equation (4.9) with respect to $T$, we have
$$-\alpha(s, T)-\sigma(s, T) \int_{s}^{T} \sigma(s, u) d u=\frac{\partial b(s, T)}{\partial T} .$$
From equations (4.10) and (4.12), it follows that
\begin{aligned} -\alpha(s, T)-\sigma(s, T) \int_{s}^{T} \sigma(s, u) d u &=\frac{\partial v(s, T)}{\partial T} \varphi_{t} \ &=-\sigma(s, T) \varphi_{t} \end{aligned}
Substituting the above into equation (4.1), we obtain
$$d f(t, T)={-\sigma(t, T) v(t, T)+\sigma(t, T) \varphi(t)} d t+\sigma(t, T) d W_{t}$$
Recall the Q-Brownian motion $Z_{t}=\int_{0}^{t} \varphi_{s} d s+W_{t}$. Substituting this into the above, we have
$$d f(t, T)=-\sigma(t, T) v(t, T) d t+\sigma(t, T) d Z_{t}$$
where the drift under $\mathbf{Q}$ is completely determined by the volatility $\sigma(t, T)$. This form is the well-known forward rate process in the HJM model. For pricing interest rate derivatives, the dynamics of the forward rates are typically simulated by equation (4.18), and the bond pricing is performed by using this form. This method is essentially the same as used with short rate models.

## 金融代写|利率建模代写Interest Rate Modeling代考|Volatility and Principal Components

This section introduces a method for constructing the volatility in the H.JM model. There are two major approaches to do so. One is a market approach; the other is a historical approach.

In the market approach, the volatility is estimated such that the model implies option prices consistent with their market prices. In the historical approach, the volatility is constructed to represent a historical dynamics of an interest rate, for example, the short rate or the forward rate. Experimentally, these two approaches result in quite different volatility structures.

When we calibrate the model for derivatives pricing, the market approach should be employed. In this, it is understood that historical volatility cannot explain market prices because the option prices are determined mostly by traders’ forecasts for the future market rather than by historical volatility. Therefore, adopting a historical approach will result in a model that misprices major derivatives. Such a model is not valid for derivatives trading.

However, when we intend to calibrate a model for interest-risk-management, the historical approach is recommended, rather than the market approach. In the historical approach, principal component analysis (PCA) is a standard technique for reducing the dimensionality of the model. PCA will be repeatedly used in this book, and we introduce the construction of volatility by applying PCA. The fundamentals of PCA and the relevant linear algebra are given in Appendix B.

## 金融代写|利率建模代写Interest Rate Modeling代考|Heath-Jarrow-Morton Framework

Let(Ω,F吨∈[0,τ],磷)是一个过滤的概率空间，其中F吨∈[0,τ]是增强过滤和磷表示真实世界的度量。到期的瞬时远期利率吨当时观察到吨表示为F(吨,吨). 用法明确时，F(吨,吨)将被称为远期利率。通常F(0,吨)表示初始远期利率。

dF(吨,吨)=一个(吨,吨)d吨+σ(吨,吨)d在吨,

σ(吨,吨)d在吨=∑l=1dσl(吨,吨)d在吨l.

## 金融代写|利率建模代写Interest Rate Modeling代考|Arbitrage Pricing and Market Price of Risk

−一个(s,吨)−σ(s,吨)∫s吨σ(s,在)d在=∂b(s,吨)∂吨.

−一个(s,吨)−σ(s,吨)∫s吨σ(s,在)d在=∂在(s,吨)∂吨披吨 =−σ(s,吨)披吨

dF(吨,吨)=−σ(吨,吨)在(吨,吨)+σ(吨,吨)披(吨)d吨+σ(吨,吨)d在吨

dF(吨,吨)=−σ(吨,吨)在(吨,吨)d吨+σ(吨,吨)d从吨

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